Graph Theory Approach to COVID-19 Transmission by Municipalities and Age Groups

The COVID-19 pandemic remains a global problem that affects the health of millions of people and the world economy. Identifying how the movement of people between regions of the world, countries, and municipalities and how the close contact between individuals of different age groups promotes the spread of infectious diseases is a pressing concern for society, during epidemic outbreaks and pandemics, such as COVID-19. Networks and Graph Theory provide adequate and powerful tools to study the spread of communicable diseases. In this work, we use Graph Theory to analyze COVID-19 transmission dynamics between municipalities of Aveiro district, in Portugal, and between different age groups, considering data from 2020 and 2021, in order to better understand the spread of this disease, as well as preparing actions for possible future pandemics. We used a digraph structure that models the transmission of SARS-CoV-2 virus between Aveiro´s municipalities and between age groups. To understand how a node fits over the contact digraphs, we studied centrality measures, namely eigencentrality, closeness, degree, and betweenness. Transmission ratios were also considered to determine whether there were certain age groups or municipals that were more responsible for the virus's spread. According to the results of this research, transmissions mostly occur within the same social groupings, that is, within the same municipalities and age groups. However, the study of centrality measures, eliminating loops, reveals that municipalities such as Aveiro, Estarreja and Ovar are relevant nodes in the transmission network of municipalities as well as the age group of 40–49 in the transmission network of age groups. Furthermore, we conclude that vaccination is effective in reducing the virus.

Stability Analysis of Delayed COVID-19 Models

We analyze mathematical models for COVID-19 with discrete time delays and vaccination. Sufficient conditions for the local stability of the endemic and disease-free equilibrium points are proved for any positive time delay. The stability results are illustrated through numerical simulations performed in MATLAB.

Analysis of a COVID-19 compartmental model: a mathematical and computational approach.

In this note, we consider a compartmental epidemic mathematical model given by a system of differential equations. We provide a complete toolkit for performing both a symbolic and numerical analysis of the spreading of COVID-19. By using the free and open-source programming language Python and the mathematical software SageMath, we contribute for the reproducibility of the mathematical analysis of the stability of the equilibrium points of epidemic models and their fitting to real data. The mathematical tools and codes can be adapted to a wide range of mathematical epidemic models.

Complex network model for COVID-19: Human behavior, pseudo-periodic solutions and multiple epidemic waves.

We propose a mathematical model for the transmission dynamics of SARS-CoV-2 in a homogeneously mixing non constant population, and generalize it to a model where the parameters are given by piecewise constant functions. This allows us to model the human behavior and the impact of public health policies on the dynamics of the curve of active infected individuals during a COVID-19 epidemic outbreak. After proving the existence and global asymptotic stability of the disease-free and endemic equilibrium points of the model with constant parameters, we consider a family of Cauchy problems, with piecewise constant parameters, and prove the existence of pseudo-oscillations between a neighborhood of the disease-free equilibrium and a neighborhood of the endemic equilibrium, in a biologically feasible region. In the context of the COVID-19 pandemic, this pseudo-periodic solutions are related to the emergence of epidemic waves. Then, to capture the impact of mobility in the dynamics of COVID-19 epidemics, we propose a complex network with six distinct regions based on COVID-19 real data from Portugal. We perform numerical simulations for the complex network model, where the objective is to determine a topology that minimizes the level of active infected individuals and the existence of topologies that are likely to worsen the level of infection. We claim that this methodology is a tool with enormous potential in the current pandemic context, and can be applied in the management of outbreaks (in regional terms) but also to manage the opening/closing of borders.

Optimal control of the COVID-19 pandemic: controlled sanitary deconfinement in Portugal.

The COVID-19 pandemic has forced policy makers to decree urgent confinements to stop a rapid and massive contagion. However, after that stage, societies are being forced to find an equilibrium between the need to reduce contagion rates and the need to reopen their economies. The experience hitherto lived has provided data on the evolution of the pandemic, in particular the population dynamics as a result of the public health measures enacted. This allows the formulation of forecasting mathematical models to anticipate the consequences of political decisions. Here we propose a model to do so and apply it to the case of Portugal. With a mathematical deterministic model, described by a system of ordinary differential equations, we fit the real evolution of COVID-19 in this country. After identification of the population readiness to follow social restrictions, by analyzing the social media, we incorporate this effect in a version of the model that allow us to check different scenarios. This is realized by considering a Monte Carlo discrete version of the previous model coupled via a complex network. Then, we apply optimal control theory to maximize the number of people returning to "normal life" and minimizing the number of active infected individuals with minimal economical costs while warranting a low level of hospitalizations. This work allows testing various scenarios of pandemic management (closure of sectors of the economy, partial/total compliance with protection measures by citizens, number of beds in intensive care units, etc.), ensuring the responsiveness of the health system, thus being a public health decision support tool.

Fractional model of COVID-19 applied to Galicia, Spain and Portugal.

A fractional compartmental mathematical model for the spread of the COVID-19 disease is proposed. Special focus has been done on the transmissibility of super-spreaders individuals. Numerical simulations are shown for data of Galicia, Spain, and Portugal. For each region, the order of the Caputo derivative takes a different value, that is not close to one, showing the relevance of considering fractional models.

A New Compartmental Epidemiological Model for COVID-19 with a Case Study of Portugal

We propose a compartmental mathematical model for the spread of the COVID-19 disease, showing its usefulness with respect to the pandemic in Portugal, from the first recorded case in the country till the end of the three states of emergency. New results include the compartmental model, described by a system of seven ordinary differential equations;proof of positivity and boundedness of solutions;investigation of equilibrium points and their stability analysis;computation of the basic reproduction number;and numerical simulations with official real data from the Portuguese health authorities. Besides completely new, the proposed model allows to describe quite well the spread of COVID-19 in Portugal, fitting simultaneously not only the number of active infected individuals but also the number of hospitalized individuals, respectively with a L2 error of 9.2152e−04 and 1.6136e−04 with respect to the initial population. Such results are very important, from a practical point of view, and far from trivial from a mathematical perspective. Moreover, the obtained value for the basic reproduction number is in agreement with the one given by the Portuguese authorities at the end of the emergency states.